[size=85] A [url=https://en.wikipedia.org/wiki/Contour_line ]contour line[/url] or lines of one potential ([url=https://en.wikipedia.org/wiki/Implicit_curve#Equipotential_curves_of_two_point_charges]equipotential lines[/url] in electrostatics) for a function of two variables is a curve at each point of which the measured quantity retains the same value. They are defined by implicit multifocal plane curves. [br] These applets construct [i]multifocal curves[/i], the geometric location of whose points is determined relative to |[b]x[/b] - [b]xᵢ[/b]| - the distances of some selected points - foci [b]Aᵢ[/b] ([b]i=1..N, Nmax=9[/b]), having given properties of preserving some selected value. By analogy with electrostatics, "[b]qi[/b] - charges" are added to the formulas as real values - weight factors to the formulas.[br] 0. Σ[b]qi[/b]*|[b]x[/b] - [b]x[sub]i[/sub][/b]|[b][sup]2[/sup][/b]=[b]const[/b] is the algebraic sum of the squares of the distances multiplied by the "charges" to the foci ([b]const[/b]:=[color=#dd7e6b][b]φ)[/b][/color]. In this case, the locus is a circle (regardless of the number and location of N points) with the center at the "[b][color=#ff0000]center of mass[/color][/b]" of this system of "charged" points and the radius calculated using the given formula. See [url=https://www.geogebra.org/m/arrctwvn]applet.[/url] [br] 1. Σ[b]qi[/b]/|[b]x[/b] - [b]x[sub]i[/sub][/b]|=[b]const[/b] -algebraic sum of inverse distances to the foci ([b]const[/b]:=[b][color=#dd7e6b]φ[/color][color=#333333]).[/color][/b] Example -[url=https://en.wikipedia.org/wiki/Implicit_curve#Equipotential_curves_of_two_point_charges][b]equipotential lines[/b][/url] in electrostatics. Images are shown in the [url=https://www.geogebra.org/m/gedefsev]applet[/url]. [br] 2. Σ[b]qi[/b]*|[b]x[/b] - [b]x[sub]i[/sub][/b]|=[b]const[/b] is the algebraic sum of the distances multiplied by the "charges" to the foci ([b]const[/b]:=[b][color=#dd7e6b]φ[/color][color=#333333])[/color][/b][color=#333333].[/color] Example - [b]ellipse[/b], [b]hyperbola[/b], [url=https://en.wikipedia.org/wiki/N-ellipse][b]Multifocal N-oval curves[/b][/url]. Images are shown in the [url=https://www.geogebra.org/m/qkeqrphu]applet[/url]. [br] 3. Π[b]qi[/b]*|[b]x[/b] - [b]x[sub]i[/sub]|[/b]=[b]const[/b] is the product of the distances to the foci: ([b]const[/b]:=[b][color=#dd7e6b]φ[/color][color=#333333])[/color][/b]multiplied by the "charges". Example -[url=https://en.wikipedia.org/wiki/Cassini_oval#Generalizations][b]N-Cassini lines[/b].[/url] Images are shown in the [url=https://www.geogebra.org/m/huuyhx7u]applet[/url]. [br] 4. Σ[b]qi[/b]*|[b]x[/b] - [b]x[sub]i[/sub][/b]|[sup][b]exponent[/b][/sup]=[b]const[/b] is the algebraic sum of the distances to some power ([b]exponent[/b]) multiplied by the "charges" to the foci ([b]const[/b]:=[b][color=#dd7e6b]φ[/color][color=#333333])[/color][/b][color=#333333].[/color][/size]

[b]Building formulas [/b][br][size=85] tx1 = If(bue1, "/", "*")[br] tx2 = If(bue1 ∨ bue2 ∨ bue2a, "Sum", "Product")[br] tx3 = If(bue2a, "^"+exponent, " ")[br] Execute[{"h(x,y)="+tx2+"(Sequence(qi(i)"+tx1+" sqrt((x - x(l0(i)))² + (y - y(l0(i)))²)"+tx3+" , i, 1,Length(l0)) )" }][/size]